Spectrum Analyzer Having a Resolution Filter that Can Be Adjusted Via Phase-Variation Parameter

ABSTRACT

A spectrum analyzer is described which comprises a mixer for mixing the conjugate complex input signal v*(t) into a base band signal x(t), and a resolution filter for narrow-band filtering the base band signal. The resolution filter has a complex, discrete impulse response using a variation parameter k 0  set to compensate for the frequency overshoot determined by the group delay of the resolution filter.

FIELD OF THE INVENTION

The invention relates to a resolution filter for a spectrum analyser.

BACKGROUND OF THE INVENTION

In spectrum analysis, a specified frequency range is “swept” with aresolution filter with a specified bandwidth. The resolution filter istherefore also sometimes referred to as a sweep filter. A resolutionfilter of this kind for a spectrum analyser with an analogous structureis known, for example, from U.S. Pat. No. 5,736,845. With resolutionfilters of the known, analogous structure, only a limited sweep rate canbe achieved; the so-called K-factor, which indicates the rate of sweep,is limited in resolution filters of the known structure.

It has been assumed hitherto in the context of spectrum analysis that itis possible to sweep within a T_(res) in the order of magnitude around1/B_(res)=T_(res), so that the resolution filter can still build up.However, this statement is only correct if a fixed filter is assumed forall the sweep rates.

A digital resolution filter for a spectrum analyser is known from DE 10105 258 A1. The resolution filter described in this context ischaracterised by a Gaussian impulse response. It is a so-calledlinear-phase resolution filter. Linear-phase resolution filters have arelatively long group delay response. As a result, during sweeping,these filters have a considerable frequency overshoot, and the middle ofthe spectrum is no longer disposed at the frequency origin. A degree ofdesign freedom, which would allow a compensation of these undesirableeffects is not provided with the impulse response of the resolutionfilter defined in DE 101 05 258 A1.

SUMMARY OF THE INVENTION

A need therefore exists for providing a spectrum analyser and anassociated resolution filter, wherein the impulse response of theresolution filter has a free design parameter, which allows acompensation of the frequency overshoot, the displacement of thefrequency origin and other undesirable effects.

According to an embodiment of the invention, the free variationparameter k₀ is introduced into the phase factor of the impulseresponse. This free variation parameter represents a degree of freedomof the phase in the design of the filter. Accordingly, for example, notonly linear-phase, but also minimal-phase filters can be realised in anefficient manner.

The free variation parameter k₀ can preferably be set in such a mannerthat the frequency overshoot determined by the group delay of theresolution filter is compensated.

As an alternative or at the same time, the variation parameter k₀ canalso be set in such a manner, that the middle of the frequency responseof the resolution filter is disposed at the frequency origin, that is tosay, at the frequency f=0.

Still other aspects, features, and advantages of the present inventionare readily apparent from the following detailed description, simply byillustrating a number of particular embodiments and implementations,including the best mode contemplated for carrying out the presentinvention. The present invention is also capable of other and differentembodiments, and its several details can be modified in various obviousrespects, all without departing from the spirit and scope of the presentinvention. Accordingly, the drawing and description are to be regardedas illustrative in nature, and not as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be explained in greater detail below with referenceto the drawings. The drawings are as follows:

FIG. 1 shows a block circuit diagram of a spectrum analyser, in whichthe resolution filter according to the invention can be used;

FIG. 2 shows a block circuit diagram of the spectrum analysis in theequivalent baseband;

FIG. 3 shows the impulse response of a linear-phase filter and of aminimal-phase Gaussian filter;

FIG. 4 shows a sweep with a minimal-phase filter and

FIG. 5 shows a sweep with a linear-phase filter.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 1 provides an overview of a spectrum analyser 20, in which theresolution filter 29 according to the invention is used. FIG. 1 showsonly the region of the signal below the intermediate frequency level,which is relevant here.

The intermediate frequency signal marked ZF in the drawings is filteredin a band-pass filter 21. The band-pass filter 21 is connected to ananalog/digital converter 22. This is followed by the I/Q mixing 23 in anI/Q demodulator 24, which, in the conventional manner, comprises a localoscillator 25 with two outputs phase-displaced through 90°, which aresupplied, together with the filtered and A/D-converted intermediatefrequency signals, to a mixer 27 of the I-branch and a mixer 26 of theQ-branch respectively.

This stage is followed by a digital filtering 28 with the resolutionfilter 29 according to the invention. An envelope-curve rectification 31then takes place in an envelope-curve rectifier 32. Logging 33 isimplemented in a logging unit 34. The logging unit 34 is connected to avideo filter 36, in which a video filtering 35 is implemented.

Various detectors 38 to 41, for example, a peak detector 38, an autopeak detector 39, a sample detector 40 and an RMS (Route Mean Square)detector may be provided in the detection stage 37. Depending on therequirements, either all four detectors can be incorporated in ahigh-performance spectrum analyser 20, or only specific detectors, forexample, only a single detector for specialised measuring tasks, needsto be installed. Evaluation and control are implemented via amicroprocessor 42.

FIG. 2 shows a simplified block circuit diagram of blocks 24, 29 and 32of the spectrum analyser 20 from FIG. 1. The complex input signal v(t)to be analysed is supplied to a complex-conjugate former 2, which formsthe complex-conjugate signal v*(t) of the input signal v(t). Thecomplex-conjugate input signal v*(t) is mixed down into the basebandsignal x(t) in a mixer 3 by multiplication with the sweep signale^(jφ(f)). The frequency f(t) of the sweep signal is shown at the top ofFIG. 2 as a function of time t, wherein it can be seen, that the sweepfrequency f(t) changes in a linear manner with time. The phase angleφ(t) as a function of time t is obtained by integration. The basebandsignal x(t) is supplied to the resolution filter 4 according to theinvention. In the resolution filter 4, the baseband signal x(t) isconvoluted with the impulse response h_(used)(t) of the resolutionfilter 4. This leads to the output signal y(t). The modulus |y(t)| ofthe signal y(t) is formed in a modulus former 5.

The lower region of FIG. 2 shows by way of example an input signal v(t),of which the spectrum consists of two discrete spectral lines. Anexample of the transmission function H(t) of the resolution filter 4 isalso shown. The spectrum shown to the right of this is displayed at theoutput of the spectrum analyser 1, wherein the spectral lines arewidened by the resolution bandwidth B_(res) of the resolution filter 4.The resolution bandwidth B_(res) corresponds to the bandwidth with anattenuation by −3 dB relative to the maximum.

By way of further explanation of the invention, the deliberations fromdocument DE 101 05 258 A1, which lead to a resolution filter with agiven impulse response, are briefly presented again below.

The spectrum of the signal v(t) is initially windowed with the impulseresponse of the resolution filter, and following this, the Fouriertransform is implemented according to the following equation:$\begin{matrix}{{S(f)} = {{\int_{- \infty}^{\infty}{{v(\tau)}{{h_{res}(\tau)} \cdot {\mathbb{e}}^{- {j\omega\tau}}}{\mathbb{d}\tau}}} = {{H_{res}(f)}*{{V(f)}.}}}} & (1)\end{matrix}$

The question of the correlation of the spectrum with white noise isrelevant in this context. This correlation describes the distance, atwhich the spectrum is un-correlated. The ACF (auto-correlation function)of the input signal with white noise is described by the followingequation: $\begin{matrix}{{E\left\{ {{v(\tau)}v*\left( {\tau + {dt}} \right)} \right\}} = {{\underset{\underset{{real}/{imag}}{︸}}{2} \cdot {N_{0}/2}}\quad{{\delta({dt})}.}}} & (2)\end{matrix}$

The ACF of the Fourier spectrum is obtained using equation (1).$\begin{matrix}{{E\left\{ {{S^{*}(f)} \cdot {S\left( {f + {df}} \right)}} \right\}} = {E\begin{Bmatrix}{\int_{- \infty}^{\infty}{{v^{*}\left( \tau_{1} \right)}{{h_{res}^{*}\left( \tau_{1} \right)} \cdot {\mathbb{e}}^{{j\omega\tau}_{1}}}{{\mathbb{d}\tau_{1}} \cdot}}} \\{\int_{- \infty}^{\infty}{{v\left( \tau_{2} \right)}{{h_{res}\left( \tau_{2} \right)} \cdot {\mathbb{e}}^{{- {j{({\omega + {d\quad\omega}})}}}\tau_{2}}}{\mathbb{d}\tau_{2}}}}\end{Bmatrix}}} \\{= {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{E\left\{ {{v^{*}\left( \tau_{1} \right)} \cdot {v\left( \tau_{2} \right)}} \right\}{{h_{res}^{*}\left( \tau_{1} \right)} \cdot}}}}} \\{{h_{res}\left( \tau_{2} \right)}{\mathbb{e}}^{{- j}\quad{\omega{({\tau_{1} - \tau_{2}})}}}{\mathbb{e}}^{{- j}\quad d\quad{\omega\tau}_{2}}{\mathbb{d}\tau_{1}}{\mathbb{d}\tau_{2}}}\end{matrix}$

Substitution of equation (2) with τ₁=τ₂:=τ leads to the followingequations: $\begin{matrix}{{E\left\{ {{S^{*}(f)} \cdot {S\left( {f + {df}} \right)}} \right\}} = {\int_{- \infty}^{\infty}{N_{0}{{h_{res}^{*}(\tau)} \cdot {h_{res}(\tau)}}{\mathbb{e}}^{{- j}\quad d\quad{\omega\tau}}{\mathbb{d}\tau}}}} \\{= {N_{0}{\int_{- \infty}^{\infty}{{{{h_{res}(\tau)}}^{2} \cdot {\mathbb{e}}^{{- j}\quad d\quad{\omega\tau}}}{\mathbb{d}\tau}}}}} \\{= {{N_{0} \cdot F}\left\{ {{h_{res}(\tau)}}^{2} \right\}}}\end{matrix}$

For a Gaussian filter, the following applies: $\begin{matrix}{{{h_{gauss}(t)} = {\sqrt{\frac{\pi}{2\quad\ln\quad(2)}}{B_{res} \cdot {\mathbb{e}}^{{- \frac{\pi^{2}}{2\quad\ln\quad{(2)}}} \cdot {(\frac{t}{T_{res}})}^{2}}}}}{{H_{gauss}(f)} = {{\mathbb{e}}^{{- 2}\quad\ln\quad{{(2)} \cdot {(\frac{f}{B_{res}})}^{2}}}.}}} & (3)\end{matrix}$

The equations below follow from equation (3): $\begin{matrix}\begin{matrix}{R_{h}(\tau)} & = & \begin{matrix}{F^{- 1}\left\{ {{H_{gauss}(f)}}^{2} \right\}} & \quad\end{matrix} & \quad \\\quad & = & {{F^{- 1}\left\{ {\mathbb{e}}^{{- 2}\quad\ln\quad{{(2)} \cdot 2 \cdot {(\frac{f}{B_{res}})}^{2}}} \right\}\quad{with}\quad B_{res}^{\prime}} = {B_{res}/\sqrt{2}}} & \quad \\\quad & = & {{{\sqrt{\frac{\pi}{2\quad\ln\quad(2)}} \cdot B_{res}^{\prime} \cdot {\mathbb{e}}^{{- \frac{\pi^{2}}{2\quad\ln\quad{(2)}}} \cdot {(\frac{\tau}{T_{res}^{\prime}})}^{2}}}{with}\quad T_{res}^{\prime}} = {T_{res}\sqrt{2}}} & \quad \\\quad & = & {\underset{\underset{:=B_{noise}}{︸}}{\sqrt{\frac{\pi}{2\quad\ln\quad(2)}} \cdot {B_{res}^{\prime}/\sqrt{2}}} \cdot {{\mathbb{e}}^{{- \frac{\pi^{2}}{2\quad\ln\quad{(2)}}} \cdot {(\frac{\tau}{\sqrt{2 \cdot}T_{res}})}^{2}}.}} & \quad\end{matrix} & (4)\end{matrix}$

Moreover, with equation (4), the equations below follow: $\begin{matrix}\begin{matrix}{{F\left\{ {{h_{gauss}(t)}}^{2} \right\}} = {{F\left\{ {\frac{\pi}{2\quad\ln\quad(2)}B_{res}^{2}{\mathbb{e}}^{{- \frac{\pi^{2}}{2\quad\ln\quad{(2)}}} \cdot 2 \cdot {(\frac{\tau}{T_{res}})}^{2}}} \right\}\quad{with}\quad T_{res}^{\prime}} = {T_{res}/\sqrt{2}}}} \\{= {{{\frac{\frac{\pi}{2\quad\ln\quad(2)}B_{res}^{2}}{\left( \frac{\pi}{2\quad\ln\quad(2)} \right)^{1/2}B_{res}^{\prime}} \cdot {\mathbb{e}}^{{- 2}\quad{{\ln{(2)}} \cdot {(\frac{f}{B_{res}^{\prime}})}^{2}}}}{with}\quad B_{res}^{\prime}} = {B_{res}\sqrt{2}}}} \\{= {\underset{\underset{B_{noise}}{︸}}{\sqrt{\frac{\pi}{2\quad\ln\quad(2)}} \cdot {B_{res}/\sqrt{2}}} \cdot {{\mathbb{e}}^{{- {\ln{(2)}}} \cdot {(\frac{f}{B_{res}})}^{2}}.}}}\end{matrix} & (5)\end{matrix}$

With a Gaussian resolution filter, the equation below is obtained usingequation (5): $\begin{matrix}{{E\left\{ {{S^{*}(f)} \cdot {S\left( {f + {df}} \right)}} \right\}} = {N_{0} \cdot B_{noise} \cdot {{\mathbb{e}}^{{- \ln}\quad{{(2)} \cdot {(\frac{df}{B_{res}})}^{2}}}.}}} & (6)\end{matrix}$

FIG. 2 shows a block circuit diagram of a spectrum analysis in theequivalent baseband. It should be borne in mind that the HF-signal v(t)to be investigated is observed in the equivalent baseband in order toallow a simpler model (i.e. without spectral components at f<0). Afterthe formation of v*(t), the signal is multiplied by the rotary phasore^(jφ(t)) leading to the following equation:x(t)=v*(t)·e ^(jφ(t))   (7)

The frequency of the rotary phasor increases in a linear manner withtime according to the following equation: $\begin{matrix}{{f(t)} = {\frac{1}{K} \cdot B_{res}^{2} \cdot {t.}}} & (8)\end{matrix}$

The K-factor indicates the sweep rate. Since the resolution filterrequires a transient time of approximately T_(res), the frequency shouldchange by a maximum of B_(res) within T_(res), which, according toequation (8), corresponds to a maximum k-factor of k=1. The followingphase is obtained by integration: $\begin{matrix}{{\varphi(t)} = {{\int_{- \infty}^{t}{2\pi\quad{f(t)}{\mathbb{d}t}}} = {\frac{\pi}{K} \cdot B_{res}^{2} \cdot {t^{2}.}}}} & (9)\end{matrix}$

The signal x(t) is then filtered through the resolution filter with theimpulse response h_(used)(t) producing the output signal y(t). Theenvelope curve |y(t)| is determined from this output signal, and thendisplayed on the spectrum analyser, generally in a logarithmic format.

The output signal is obtained from the following equation:y(t) = x(t)^(*)h_(used)(t) = ∫_(−∞)^(∞)h_(used)(τ) ⋅ x(t − τ)  𝕕τ

Introducing equation (7) leads to the following equation:y(t) = ∫_(−∞)^(∞)h_(used)(τ) ⋅ v^(*)(t − τ)𝕖^(jφ(t − τ))  𝕕τ

Introducing equation (9) finally leads to the following equation:${y(t)} = {\int_{- \infty}^{\infty}{{{h_{used}(\tau)} \cdot {v^{*}\left( {t - \tau} \right)}}{\mathbb{e}}^{j{\frac{\pi}{K} \cdot B_{res}^{2} \cdot {({t - \tau})}^{2}}}\quad{\mathbb{d}\tau}}}$

The following equation is obtained by multiplying out: $\begin{matrix}\begin{matrix}{{y(t)} = {\underset{\underset{{\mathbb{e}}^{{j\varphi}{(1)}}}{︸}}{{\mathbb{e}}^{j{\frac{\pi}{K} \cdot B_{res}^{2} \cdot t^{2}}}}{\int_{- \infty}^{\infty}{\underset{\underset{h_{disp}(\tau)}{︸}}{{h_{used}(\tau)}{\mathbb{e}}^{j{\frac{\pi}{K} \cdot B_{res}^{2} \cdot \tau^{2}}}} \cdot}}}} \\{{{v^{*}\left( {t - \tau} \right)}{\mathbb{e}}^{{- j}{\frac{2\pi}{K} \cdot B_{res}^{2} \cdot t}\quad\tau}{\mathbb{d}\tau}},} \\\quad\end{matrix} & (10)\end{matrix}$

wherein the first term e^(jφ(t)) is not problematic, because ultimately|y(t)| will be displayed. The impulse response: $\begin{matrix}{{h_{disp}(t)} = {{h_{used}(t)} \cdot {\mathbb{e}}^{j{\frac{\pi}{K} \cdot B_{res}^{2}}t^{2}}}} & (11)\end{matrix}$

is introduced into the equation. The index “disp” denotes “displayed”,because the spectrum of this impulse response is displayed, as will beshown below.

According to equation (8), the following equation is obtained byconversion: $\begin{matrix}{t = {\frac{{f(t)} \cdot K}{B_{res}^{2}}.}} & (12)\end{matrix}$

The following equation is obtained by substitution into equation (10):$\begin{matrix}{{y(t)} = {{\mathbb{e}}^{{j\varphi}{(t)}}{\int_{- \infty}^{\infty}{{{h_{disp}(\tau)} \cdot {v^{*}\left( {t - \tau} \right)}}{\mathbb{e}}^{{- {{j\omega}{(t)}}} \cdot \tau}\quad{{\mathbb{d}\tau}.}}}}} & (13)\end{matrix}$

Now, some interesting statements can be made: comparing equation (13)with the Fourier analysis in equation (1) shows that

1. The “displayed” resolution filter h_(disp)(t) described according toequation (11) rather than the used resolution filter h_(used)(t) isdisplayed in the spectrum analysis. With a slow sweep for approximatelyk≧2, h_(used)(t) and h_(disp)(t) are approximately in agreement.However, with a faster sweep, significant differences occur. In thiscase, the level collapses, and the displayed resolution filter becomesbroader (the filter can no longer build up).

2. In equation (13), by contrast with the Fourier analysis, the timesignal displaced by t is used rather than v(τ). As a result, thespectrum analyser evaluates an observation interval sliding over time,which is not in itself problematic. However, the question regarding theinfluence of the rate of the sliding observation window on the outputspectrum is particularly relevant.

In order to evaluate the question of the sliding observation window inpoint 2 in an improved manner, it is advisable to apply the Parseval'stheorem, as set out below, to equation (13):∫_(−∞)^(∞)x₁(τ) ⋅ x₂^(*)(τ)𝕕τ = ∫_(−∞)^(∞)X₁(F) ⋅ X₂^(*)(F)𝕕F

By substitution of:${x_{1}(\tau)} = {{{{h_{disp}(\tau)} \cdot {\mathbb{e}}^{{- {{j\omega}{(t)}}}\tau}}\overset{\tau}{->}{X_{1}(F)}} = {H_{disp}\left( {F + {f(t)}} \right)}}$${x_{2}(\tau)} = {{{v\left( {t - \tau} \right)}\overset{\tau}{->}{X_{2}(F)}} = {{V\left( {- F} \right)} \cdot {\mathbb{e}}^{{j2\pi}\quad{Ft}}}}$

equation (13) can be described by the following equations:$\begin{matrix}\begin{matrix}{{y(t)} = {{\mathbb{e}}^{{j\varphi}{(t)}}{\int_{- \infty}^{\infty}{{{H_{disp}\left( {F + {f(t)}} \right)} \cdot {V^{*}\left( {- F} \right)}}{\mathbb{e}}^{{- {j2\pi}}\quad{Ft}}\quad{\mathbb{d}F}}}}} \\{= {{\mathbb{e}}^{{j\varphi}{(t)}}{\int_{- \infty}^{\infty}{{{H_{disp}\left( {F - {f(t)}} \right)} \cdot {V^{*}(F)}}{\mathbb{e}}^{{- {j2\pi}}\quad{Ft}}{{\mathbb{d}F}.}}}}} \\\quad\end{matrix} & (14)\end{matrix}$

Accordingly, as anticipated, a convolution of the input spectrum withthe resolution filter is obtained according to the following equation:y(t)=e ^(jφ(t)) H _(disp)(f(t))*[V*(f(t))e ^(−j2πf(t)t)]

By inserting equation (12) into equation (14), the following equation isfinally obtained: $\begin{matrix}{{y(t)} = {{\mathbb{e}}^{{j\varphi}\quad{(t)}}{\int_{- \infty}^{\infty}{{{H_{disp}\left( {F - {f(t)}} \right)} \cdot V}*(F){\mathbb{e}}^{{- j}\frac{2\pi\quad K}{B_{res}^{2}}{{Ff}{(t)}}}\quad{{\mathbb{d}F}.}}}}} & (15)\end{matrix}$

DE 101 05 258 A1 derives only a sweep-optimised Gaussian filter. Thissweep-optimised Gaussian filter must be a linear-phase filter.

New insights have shown that the sweep-optimised filter can be derivedfor a random filter. The given filter can be random both with regard toits magnitude and also with regard to phase. It is particularly relevantthat no restrictions are placed on phase. In the case of the sweepfilter known from DE 101 05 258 A1, this degree of freedom could not beexploited, because the linear-phase properties were required in thatcontext. With a randomly-specifiable phase, minimal-phase filters cannow be realised according to the invention, which are optimised withregard to the necessary transient time.

Equation (10) defines the “displayed” impulse response h_(disp)(t) andthe “used” impulse response h_(used)(t). In this context, the impulseresponse h_(disp)(t) describes the transform of the frequency responseH_(disp)(f) of the resolution filter displayed in the sweep, while theimpulse response h_(used)(t) is the transform of the used filter withthe frequency response H_(used)(f). The correlation between the twoimpulse responses:${h_{disp}(t)} = {{h_{used}(t)} \cdot {\mathbb{e}}^{j\frac{\pi}{K}B_{res}^{2}t^{2}}}$

is already known from equation (11). Now, the procedure according to theinvention consists in developing the frequency response of the displayedfilter H_(disp)(f) rather than the frequency response H_(used)(f) of theused filter. Since only the modulus frequency response is displayed inspectrum analysis, the phase can be selected at random. According tothis design, the impulse response h_(disp)(t) is calculated byre-transformation. In the next stage, the sought impulse responseh_(used)(t) of the sweep-optimised filter is calculated according to theabove formula with the sweep rate k.

The individual design stages are described in greater detail below:

1. Specification of the desired displayed modulus frequency response|H_(disp)(f)|:

A Gaussian filter is often used. However, filters with a lesssteeply-declining modulus frequency response can also be of interest,because this can reduce the group delay response. This is associatedwith a shorter transient time, which is particularly desirable in thecase of applications with frequent transient processes of the filter.

2. Specification of the phase of H_(disp)(f):

In principle, the phase can be specified randomly. In order to achieve aminimal group delay response, it is advisable to use a minimal-phasefilter. Details of the design are given below. In this context, it isassumed that the filter is realised as a digital filter, that is to say,the discrete impulse response h_(disp)(k) with k=[0,nof_(Taps)−1] iscalculated, wherein nof_(Taps) describes the number of taps.

The simplest case is a design of a linear-phase filter with a lengthnof_(Taps) with a specified modulus frequency response |H_(disp)(f)|.Currently available methods include the Remez algorithm and the MMSalgorithm. Following this, the zero positions of the transmissionfunction in the complex z-plane are determined. The zero positionsoutside the circle of radius 1 are then reflected into the circle ofradius 1, so that the modulus frequency response remains unchanged. Thismethod is sub-optimum, because it always leads to minimal-phase filterswith doubled zero positions, which limits the degree of freedom.

FIG. 3 shows an example of the discrete impulse response h(k) as afunction of the sampling index of a minimal-phase Gaussian filter bycomparison with a linear-phase Gaussian filter. It is evident that theminimal-phase filter has a substantially shorter group delay responsethan the linear-phase filter. Of course, the minimal-phase filter hasthe same number of taps nof_(Taps)=161 as the linear-phase filter, thatis to say, the transient time is of the same length. However, during thesweep, the transient error of the minimal-phase filter decayssubstantially more rapidly than in the case of the linear-phase filter,as can be seen from a comparison of FIG. 4 with FIG. 5. FIG. 4 shows thesweep with a minimal-phase filter, while FIG. 5 shows the same sweepwith a linear-phase filter. In each case, the input signal is a discretespectral line. It is evident that, in the case of the minimal-phasefilter, the error has already decayed so severely during the transientphase, i.e. immediately after the “first lobe”, that approximately thesecond half of the transient phase can be used for the analysis. Bycontrast, with the linear-phase filter, the analysis cannot be starteduntil later.

An improved filter design is achieved with the following method. Thelinear-phase filter is not designed with nof_(Taps) taps, but ratherwith double the length 2·nof_(Taps). As a target function,|H_(disp)(f)|² is also specified rather than |H_(disp)(f)|. The sameapplies for any cost functions used. Following this, the zero positionsin the circle of unit radius are calculated from the determined digitalfilter h_(disp) ^((long))(k). The mirror-image symmetrical zeropositions outside the circle of unit radius are rejected. The impulseresponse h_(disp)(k) generated in this manner has the desired number oftaps nof_(Taps). Moreover, it also has the desired target modulusfrequency response |H_(disp)(f)|. The filter calculated in this mannerhas no doubled zero positions, so that this design exploits the fulldegree of freedom.

3. Calculation of h_(used)(k):

The impulse response h_(disp)(k) is now available. According to equation(11), the impulse response of the filter to be used is calculated usingthe following equation: $\begin{matrix}{{{h_{used}(k)} = {{{{h_{disp}(k)} \cdot {\mathbb{e}}^{{{- j}\frac{\pi}{K}{{B_{res}^{2}{({k - k_{0}})}}^{2} \cdot T_{a}^{2}}}\quad}}\quad{with}{\quad\quad}k} = \left\lbrack {0,{{{nof}\text{“""”}_{Taps}} - 1}} \right\rbrack}},} & (16)\end{matrix}$

wherein T_(a) is the sampling period of the digital filter. By contrastwith equation (11), the parameter k₀, which brings about a displacementof the spectrum within the spectral range, has been introduced accordingto the invention. Various desirable effects can be achieved with theparameter k₀:

1. Compensation of the frequency overshoot: as a result of the groupdelay of the resolution filter, the corresponding frequency of theoutput signal has a similar overshoot. This effect can be compensated bysetting an appropriate k₀.

2. Minimisation of the required bandwidth of H_(used)(f): with anincreasing sweep rate (smaller k), the middle of the spectrumH_(used)(f) is no longer at the frequency origin f=0, but is displacedtowards the higher frequencies. With the present digital system, therequired sampling rate f_(a) would therefore have to be unnecessarilyincreased, in order to continue to fulfil the sampling theorem. Thiseffect can be avoided by an appropriate choice of k₀. In a goodapproximation, k₀=k_(max) should be selected, k_(max) being the time ofthe maximum of |h_(disp)(k)|.

Reference must also be made to a procedure, which is meaningful for theimplementation: h_(used)(k) should never be pre-calculated for differentsweep rates k and stored in the device; only h_(disp)(k) should bepre-calculated. The relevant impulse response h_(used)(k) will becalculated in the device according to the specifications in equation(16) after the desired sweep rate is known. This method has theadvantage that the housekeeping and memory requirement for theindividual impulse responses does not occur. Furthermore, the impulseresponse for the present k is always used. As a result, no quantisationerrors occur, because the impulse response h_(used)(k) for the exactk-factor set is not available.

The closed display of the impulse response of the complex resolutionfilter is determined as follows.

For the derivation of the closed display of the impulse responseh_(used) of the filter, the time-continuous case will first beconsidered. In this case, the free variation parameter is t₀=k₀·T_(a).

A Gaussian resolution filter with the bandwidth B_(res) is used for thespectrum analysis. The “displayed” resolution filter should have thefollowing impulse response and transmission function: $\begin{matrix}{{{h_{disp}(t)} = {\sqrt{\frac{\pi}{2{\ln(2)}}}{B_{res} \cdot {\mathbb{e}}^{{- \frac{\pi^{2}}{2{\ln{(2)}}}} \cdot {(\frac{t}{T_{res}})}^{2}}}*{h_{allp}(t)}}}{{H_{disp}(f)} = {{\mathbb{e}}^{{- 2}{{\ln{(2)}} \cdot {(\frac{f}{B_{res}})}^{2}}} \cdot {\mathbb{e}}^{{j\varphi}{(f)}}}}} & (17)\end{matrix}$

The amplitude-corrected display of the spectral lines is secured with|H_(disp)(f=0)|=1. The phase characteristic φ(f) of the transmissionfunction can be selected randomly, because |H_(disp)(f)| is displayed inthe spectrum analysis. The phase response φ(f) can be determined, forexample, according to the method described above, so that thetransmission function is a minimal-phase function, which leads to arapid transient of h_(disp)(t). The Fourier retransform of e^(jφ(f)) ish_(allp)(t). Since a multiplication with e^(jφ(f)) is implemented in thefrequency domain, a corresponding convolution with the impulse responseh_(allp)(t) of an all-pass filter must be implemented in the timedomain, which is symbolised by the character *. Because of thecorrelation indicated in equation (11) between h_(disp)(t) andh_(used)(t), φ(t) describes not only the phase response of thetransmission function H_(disp)(f), but also the phase response of thetransmission function H_(used)(f) of the resolution filter.

The impulse response of the resolution filter to be used is obtainedaccording to equation (16) in a time-continuous display as:$\begin{matrix}{{h_{used}(t)} = {{h_{disp}(t)} \cdot {{\mathbb{e}}^{{- j}\frac{\pi}{\quad K}\quad{B_{\quad{res}}^{\quad 2}{({t\quad - \quad t_{\quad 0}})}}^{2}}.}}} & (18)\end{matrix}$

The following equation is obtained by substitution: $\begin{matrix}{{h_{used}(t)} = {\sqrt{\frac{\pi}{2{\ln(2)}}}{B_{res} \cdot \left\lbrack {{\mathbb{e}}^{{- \frac{\pi^{2}}{2{\ln{(2)}}}} \cdot {(\frac{t}{T_{res}})}^{2}}*{h_{allp}(t)}} \right\rbrack \cdot {{\mathbb{e}}^{{- j}\frac{\pi}{K}{B_{res}^{2}{({t - t_{0}})}}^{2}}.}}}} & (19)\end{matrix}$

The transition to the discrete impulse response follows from:h _(used)(k)=T _(a) h _(used)(t=kT _(a))

By substitution of equation (19), the following equation is obtained:$\begin{matrix}{{h_{used}(k)} = {\sqrt{\frac{\pi}{2{\ln(2)}}} \cdot B_{res} \cdot T_{a} \cdot {\quad{{\left\lbrack {{\mathbb{e}}^{{- \frac{\pi^{2}}{2{\ln{(2)}}}} \cdot {(\frac{t}{T_{res}})}^{2}}*{h_{allp}(t)}} \right\rbrack \cdot {\mathbb{e}}^{{- j}\frac{\pi}{K}{B_{res}^{2}{({t - t_{0}})}}^{2}}}❘_{\begin{matrix}{t_{0} = {k_{0} \cdot T_{a}}} \\{t = {kT}_{a}}\end{matrix}}}}}} & (20)\end{matrix}$

With T_(res)=1/B_(res), B_(res)=resolution bandwidth with a 3 dB signaldecay relative to the maximum and T_(a)=sampling period in the baseband.

The following general presentation for the impulse response is obtained:h _(used)(k)=C ₁ ·[e ^(−C) ² ^(T) ^(a) ² ^(·k) ² *h _(allp)(t)]·e ^(−jC)³ ^((k−k) ⁰ ⁾ ² ^(·T) ^(a) ²

In this equation, k denotes the sampling index, and T_(a) denotes thesampling period. C₁, C₂ and C₃ are constants, wherein the value of theconstant C₁ is preferably:$C_{1} = {\sqrt{\frac{\pi}{2{\ln(2)}} \cdot}{B_{res} \cdot T_{a}}}$

and wherein B_(res) is the bandwidth of the resolution filter.

The value of the constant C₂ is preferably:${C_{2} = {\frac{\pi^{2}}{2{\ln(2)}} \cdot \frac{1}{T_{res}^{2}}}},$

wherein T_(res)=1/B_(res) is the reciprocal bandwidth B_(res) of theresolution filter.

The value of the constant C₃ is preferably${C_{3} = {\frac{\pi}{K} \cdot B_{res}^{2}}},$

wherein B_(res) is the bandwidth of the resolution filter, and K is thek-factor of the resolution filter, which is defined by the equation:${f(t)} = {\frac{1}{K} \cdot B_{res}^{2} \cdot t}$

and f(t) is a linear frequency variable with time t, which is suppliedto the mixer 3 of the spectrum analyser connected upstream of theresolution filter 4.

However, the constants C₁, C₂ and C₃ may also conceivably be specifiedin a different manner within the framework of the present invention.

While the present invention has been described in connection with anumber of embodiments and implementations, the present invention is notso limited but covers various obvious modifications and equivalentarrangements, which fall within the purview of the appended claims.

1. A resolution filter for a spectrum analyser, wherein the resolutionfilter has the following complex, discrete impulse response h_(used)(k):h _(used)(k)=C ₁ ·[e ^(−C) ² ^(T) ^(a) ² ^(·k) ² *h _(allp)(t)]·e ^(−jC)³ ^((k−k) ⁰ ⁾ ² ^(·T) ^(a) ² wherein C₁, C₂ and C₃ are constants, k isthe sampling index and T_(a) is the sampling period, wherein h_(allp)(t)is the Fourier retransform of e^(jφ(f)), in which φ(f) is arandomly-specified phase response dependent upon the frequency of thetransmission function of the resolution filter, wherein k₀ is a freevariation parameter and wherein the variation parameter k₀ is set insuch a manner that the frequency overshoot determined by the group delayof the resolution filter is compensated.
 2. A resolution filteraccording to claim 1, wherein: the variation parameter k₀ is set in sucha manner that the middle of the frequency response H_(used)(f) of theresolution filter is disposed at the frequency origin at the frequencyf=0.
 3. A resolution filter according to claim 1, wherein: φ(f) andh_(allp)(t) are selected in such a manner that a minimal-phaseresolution filter is formed.
 4. A resolution filter according to claim1, wherein: the value of the constant C₁ is:$C_{1} = {\sqrt{\frac{\pi}{21{n(2)}}} \cdot B_{res} \cdot T_{a}}$wherein B_(res) is the bandwidth of the resolution filter.
 5. Aresolution filter according to claim 1, wherein: the value of theconstant C₂ is${C_{2} = {\frac{\pi^{2}}{21{n(2)}} \cdot \frac{1}{T_{res}^{2}}}},$wherein T_(res)=1/B_(res) is the reciprocal bandwidth B_(res) of theresolution filter.
 6. A resolution filter according to claim 1, wherein:the value of the constant C₃ is${C_{3} = {\frac{\pi}{K} \cdot B_{res}^{2}}},$ wherein B_(res) is thebandwidth of the resolution filter and K is the K-factor of theresolution filter, wherein the K-factor is defined via the equation:${f(t)} = {\frac{1}{K} \cdot B_{res}^{2} \cdot t}$ and f(t) is afrequency variable with time t in a linear manner, which is supplied toa mixer of the spectrum analyser connected upstream of the resolutionfilter.
 7. A spectrum analyser for analysing the spectrum of an inputsignal with a resolution filter specifying the frequency resolution,wherein the resolution filter has the following complex, discreteimpulse response h_(used)(k):h _(used)(k)=C ₁ ·[e ^(−C) ² ^(T) ^(a) ² ^(·k) ² *h _(allp)(t)]·e ^(−jC)³ ^((k−k) ⁰ ⁾ ² ^(·T) ^(a) ² wherein C₁, C₂ and C₃ are constants, k isthe sampling index and T_(a) is the sampling period, wherein h_(allp)(t)is the Fourier retransform of e^(jφ(f)), in which φ(f) is arandomly-specified phase response dependent upon the frequency of thetransmission function of the resolution filter, wherein k₀ is a freevariation parameter and wherein the variation parameter k₀ is set insuch a manner that the frequency overshoot determined by the group delayof the resolution filter is compensated.
 8. A spectrum analyseraccording to claim 7, wherein: the variation parameter k₀ is set in sucha manner that the middle of the frequency response H_(used)(f) of theresolution filter is disposed at the frequency origin at the frequencyf=0.
 9. A spectrum analyser according to claim 7, wherein: φ(f) andh_(allp)(t) are selected in such a manner that a minimal-phaseresolution filter is formed.